Geometry 1 Unit 6

Quadrilaterals

Geometry 1 Unit 6

6.1 Polygons

Polygons Polygon

A closed figure in a plane Formed by connecting line segments endpoint to

endpoint Each segment intersects exactly 2 others Classified by the number of sides they have Named by listing vertices in consecutive order

Sides Line segments in a polygon

Vertex Each endpoint in a polygon

Polygons

polygons not polygons

Polygons

Pentagon ABCDE or pentagon CDEAB

A

D C

BE

PolygonsSides Name3 Triangle4 Quadrilateral5 Pentagon6 Hexagon7 Heptagon8 Octagon9 Nonagon10 Decagon11 Undecagon12 Dodecagonn n-gon (a 19 sided polygon is a 19-gon)

Polygons

Diagonals Line segments that connect non-

consecutive vertices.

Polygons

Convex polygons Polygons with no diagonals on the

outside of the polygon

Polygons

Concave polygons A polygon is concave if at least one

diagonal is outside the polygon These are also called nonconvex.

Polygons

Example 1 Identify the polygon and state whether it

is convex or concave.

Polygons

Equilateral Polygon all sides the same length

Equiangular Polygon all angles equal measure

Regular Polygon equilateral and equiangular

Polygons

Equilateral Equiangular Regular

Polygons

Example 2 Decide whether the polygon is regular.

Polygons

Interior Angles of a Quadrilateral Theorem The sum of the measures of the interior

angles of a quadrilateral is 360°.

m1 + m2 + m3 + m4 = 360°

1

4 3

2

Polygons

Example 3 Find mF, mG, and mH.

x

x55°E

H

F

G

Polygons

Example 4 Use the information in the diagram to

solve for x

100°

2x + 30 3x – 5

120°

Geometry 1 Unit 6

6.2 Properties of Parallelograms

Properties of Parallelograms

ParallelogramQuadrilateral with two pairs of parallel

sides.

Properties of Parallelograms

Opposite Sides of a Parallelogram Theorem If a quadrilateral is a parallelogram, then

its opposite sides are congruent.

P

Q

S

RPQ RS and SP QR

Properties of Parallelograms

Opposite Angles in a Parallelogram Theorem If a quadrilateral is a parallelogram, then

its opposite angles are congruent.

P

Q

S

RP R and Q S

Properties of Parallelograms

Consecutive Angles in a Parallelogram Theorem If a quadrilateral is a parallelogram, then

its consecutive angles are supplementary

Add to equal 180°

P

Q

S

R

mP + mQ = 180°

mQ + mR = 180°

mR + mS = 180°

mS + mP = 180°

Properties of Parallelograms

Diagonals in a Parallelogram Theorem If a quadrilateral is a parallelogram, then

its diagonals bisect each other.

P

Q

S

R

M

QM SM and PM RM

Properties of Parallelograms

Example 1 GHJK is a parallelogram. Find each

unknown length JH LH

G

K

H

J

L

68

Properties of Parallelograms

Example 2 In ABCD, mC = 105°. Find the

measure of each angle. mA mD

Properties of Parallelograms

Example 3 WXYZ is a parallelogram. Find the value

of x.

W

YZ

X

3x + 18°

4x – 9°

Properties of Parallelograms Example 4

Given: ABCD is a

parallelogram. Prove:

2 4

Statement Reasons

ABCD is a parallelogram

AD || BC

2 1

AB || CD

Alternate interior angles theorem

2 4

A

D

B

C

1

4

2

3

Properties of Parallelograms Example 5

Given: ACDF is a

parallelogram. ABDE is a

parallelogram. Prove:

∆BCD ∆EFA

Statement Reason

ACDF is a parallelogram.

ABDE is a parallelogram.

Opposite sides of a parallelogram are congruent

AC = DF

AB = DE

AC = AB + BC

DF = DE + EF

AC = DE + DF

AB + BC = AB + EF

BC = EF

Def of Congruent

∆BCD ∆EFA

A B C

F DE

Properties of Parallelograms

Example 6 A four-sided concrete slab has

consecutive angle measures of 85°, 94°, 85°, and 96°. Is the slab a parallelogram? Explain.

Geometry 1 Unit 6

6.3 Proving Quadrilaterals are Parallelograms

Proving Quadrilaterals are Parallelograms

Investigating Properties of Parallelograms Cut 4 straws to form two congruent pairs. Partly unbend two paperclips, link their smaller

ends, and insert the larger ends into two cut straws. Join the rest of the straws to form a quadrilateral with opposite sides congruent.

Change the angles of your quadrilateral. Is your quadrilateral a parallelogram?

Proving Quadrilaterals are Parallelograms

Converse of the Opposite Sides of a Parallelogram Theorem If a opposite sides of a quadrilateral are

congruent, then the quadrilateral is a parallelogram.

D

A

C

BABCD is a parallelogram

Proving Quadrilaterals are Parallelograms

Converse of the Opposite Angles in a Parallelogram Theorem If both pairs of opposite angles of a

quadrilateral are congruent, then the quadrilateral is a parallelogram.

D

A

C

B ABCD is a parallelogram.

Proving Quadrilaterals are Parallelograms

Converse of the Consecutive Angles in a Parallelogram Theorem If an angle of a quadrilateral is supplementary to both

of its consecutive angles, then the quadrilateral is a parallelogram.

ABCD is a parallelogram

D

A

C

B

(180 – x)°

x°

x°

Proving Quadrilaterals are Parallelograms

Converse of the Diagonals in a Parallelogram Theorem If the diagonals of a quadrilateral bisect each

other, then the quadrilateral is a parallelogram.

D

A

C

B

M

ABCD is a parallelogram

Proving Quadrilaterals are Parallelograms

Example 1 Given:

∆PQT ∆RST

Prove: PQRS is a

parallelogram.

Statements Reasons

∆PQT ∆RST

CPCTC

PT = RT

ST = QT

Def. of bisect

PQRS is a parallelogram

P Q

RS

T

Proving Quadrilaterals are Parallelograms

Example 2 A gate is braced as shown. How do you

know that opposite sides of the gate are congruent?

Proving Quadrilaterals are Parallelograms

Congruent and Parallel Sides Theorem If one pair of opposite sides of a

quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.

A

B

D

C

ABCD is a parallelogram.

Proving Quadrilaterals are Parallelograms

To determine if a quadrilateral is a parallelogram, you need to know one of the following: Opposite sides are parallel Opposite sides are congruent Opposite angles are congruent An angle is supplementary with both of its

consecutive angles Diagonals bisect each other One pair of sides is both parallel and congruent

Proving Quadrilaterals are Parallelograms

Example 3 Show that A(-1,2), B(3,2), C(1,-2), and

D(-3,-2) are the vertices of a parallelogram.

Geometry 1 Unit 6

6.4 Rhombuses, Rectangles, and Squares

Rhombuses, Rectangles, and Squares

RectangleParallelogram with four congruent angles

RhombusParallelogram with four congruent sides

SquareParallelogram with four congruent angles

and four congruent sides

Rhombuses, Rectangles, and Squares

Example 1Decide if each statement is always,

sometimes or never true.A rhombus is a rectangle

A parallelogram is a rectangle

A rectangle is a square

A square is a rhombus

Rhombuses, Rectangles, and Squares

Example 2Given FROG is a rectangle, what else do

you know about FROG?

F R

G O

Rhombuses, Rectangles, and Squares

Example 3EFGH is a rectangle. K is the midpoint of

FH. EG = 8z – 16, What is the measure of segment EK?What is the measure of segment GK?

Rhombuses, Rectangles, and Squares

Rhombus CorollaryA quadrilateral is a rhombus if and only if

it has four congruent sides.

Rhombuses, Rectangles, and Squares

Rectangle CorollaryA quadrilateral is a rectangle if and only if

it has four right angles.

Rhombuses, Rectangles, and Squares

Square CorollaryA quadrilateral is a square if and only if it

is a rhombus and a rectangle.

Rhombuses, Rectangles, and Squares

Perpendicular Diagonals of a Rhombus TheoremA parallelogram is a rhombus if and only if

its diagonals are perpendicular.

B

A

C

D

ABCD is a rhombus if and only if

AC BD.

Rhombuses, Rectangles, and Squares

Diagonals Bisecting Opposite Angles Theorem. A parallelogram is a rhombus if and only

if each diagonal bisects a pair of opposite angles.

A

B

D

C ABCD us a rhombus if and only if

AC bisects DAB and BCD

and

BD bisects ADC and CBA

Rhombuses, Rectangles, and Squares

Diagonals in a Rectangle Theorem A parallelogram is a rectangle if and only

if its diagonals are congruent.

A B

D C

ABCD is a rectangle if and only if

AC BD.

Rhombuses, Rectangles, and Squares

Example 4 You cut out a parallelogram shaped quilt

piece and measure the diagonals to be congruent. What is the shape?

An angle formed by the diagonals of the quilt piece measures 90°. Is the shape a square?

Geometry 1 Unit 6

6.5 Trapezoids and Kites

Trapezoids and Kites

Trapezoid A quadrilateral with exactly one pair of parallel sides.

Bases The parallel sides of a trapezoid.

Pairs of Base Angles Angles in a trapezoid that share a base.

Legs The nonparallel sides of a trapezoid.

Isosceles Trapezoid Trapezoid with congruent legs.

Trapezoids and Kites

Base Angles of an Isosceles Trapezoid Theorem If a trapezoid is isosceles, then each pair

of base angles is congruent.

A B, C D

D

A B

C

Trapezoids and Kites

Congruent Base Angles in a Trapezoid Theorem. If a trapezoid has a pair of congruent

base angles, then it is an isosceles trapezoid.

D

A B

C

ABCD is an isosceles trapezoid.

Trapezoids and Kites

Diagonals in an Isosceles Trapezoid Theorem A trapezoid is isosceles if and only if its

diagonals are congruent.

D

A B

C

ABCD is isosceles if and only if

AC BD.

Trapezoids and Kites

Midsegment of a trapezoid The segment that connects the

midpoints of a trapezoids legs.

midsegment

Trapezoids and Kites

Midsegment Theorem for Trapezoids The midsegment of a trapezoid is parallel

to each base and its length is one half the sum of the lengths of the bases.

B C

D

NM

A

MN || AD, MN || BC,

MN = ½(AD + BC)

Trapezoids and Kites

Kite A quadrilateral with two distinct pairs of

consecutive congruent sides. Opposite sides are not congruent.

Diagonals of a Kite Theorem If a quadrilateral is a kite, then its

diagonals are perpendicular.

Opposite Angles in a Kite Theorem If a quadrilateral is a kite, then exactly

one pair of opposite angles are congruent.

Trapezoids and Kites

Example 4 GHJK is a kite. Find HP.

5

√29

G

H

PJ

K

Trapezoids and Kites

Example 5 RSTU is a kite. Find mR, mS, and

mT.

x + 30°

125°

x°R

S

T

U

Geometry 1 Unit 6

6.6 Special Quadrilaterals

Special Quadrilaterals

Special Quadrilaterals

Property Parallelogram Rectangle Rhombus Square Trapezoid Kite

1.Both pairs of opposite sides are congruent

2. Diagonals are congruent

3. Diagonals are perpendicular

Special QuadrilateralsProperty Parallelogra

mRectangle Rhombus Square Trapezoid Kite

4. Diagonals bisect each other

5. Consecutive angles are supplementary

6. Both pairs of opposite angles are congruent

Geometry 1 Unit 6

6.7 Areas of Triangles and Quadrilaterals

Area

Area is the number of square units in a figure.

Area-Rectangles

Count the number of squares to find the area.

A shortcut is to find the length and multiply it by the width.

Area-Parallelograms

Count the number of squares to find the area.

A shortcut is to find the length and multiply it by the width.

Rectangles and Parallelograms

Area = length times width Area = base times height A = bh

b b

h

Area-Triangles

Area-Triangles

A = ½ base times height A = ½ bh

hh

h

b

b

b

Area-Trapezoids

b2

h

b1

hb1

h

b2

A = triangle 1 + triangle 2

A = ½ h(b1) + ½ h(b2)

A = ½ h(b1 + b2)

Area-Trapezoids

A = ½ height times (base1 + base 2) A = ½ h(b1 + b2)

h h h

b2 b2 b2

b1 b1b1

Area- Kites

A = ½ (diagonal 1) times (diagonal 2) A = ½ (d1)(d2)

d1 and d2

Area-Rhombus

A = ½ (diagonal 1) times (diagonal 2) A = ½ (d1)(d2)

d1 and d2

Example 1

Find the area of ΔRST.

3

4 TS

R

Example 2

What is the base of a triangle that has an area of 48 and a height of 3?

Example 3

A rectangle has an area of 100 square meters and a height of 25 meters. Are all the rectangles with these dimensions congruent?

Example 4

Find the area of parallelogram RSTU.

U

R

6

6

3

S

T

Example 5

What is the height of a parallelogram that has an area of 96 square feet and a base length of 8 feet?

Example 6

Find the area of trapezoid EFGH.E(-2, 3), F(2, 4), G(2, -2), H(-2, -1)

Example 7

Find the area of kite ABCD.A(0, 5), B(3, 6), C(6, 5), D(3, 2)

Example 8

Use the information given in the diagram to find the area of kite ABCD.

A

D

C

B

8

8

8

16

Example 9

The tray below is designed to save space on cafeteria tables. How much table area does the tray use?

16 in

8 in

3 in

10 in

Example 10

Find the area of rhombus EFGH if EG = 10 and FH = 15.